For the sake of better analyzing the existence of a cut on the
negative real axis, we also used the so-called D-Log Padé
approximants [1]}, [2]}. They are a suitable
alternative for functions with branch cuts because, instead of working with
the original function \(f(z)\) , which has a cut, one tries to
approximate a new function \(F(z)\) , which only has simple poles, and
then unfold the procedure. In particular, let us assume that the
function we are interested in is given
by [1]}, [2]}
\(f(z) = A(z) \, \frac{1}{(\mu - z)^\gamma } + B(z)\) ,
where \(A(z)\)  and \(B(z)\)  have a simple structure and are analytic at \(z=\mu \) ,
and where \(\gamma \)  can be any real number. We then construct a new
function defined as [1]}, [2]}
\(F(z) = \frac{\mathrm {d}}{\mathrm {d} z}\ln {f(z)} \approx \frac{\gamma }{(\mu - z)}\) . Thus,
the branch point \(\mu \)  of \(f(z)\)  turns into a simple pole in \(F(z)\) ,
whose residue is equal to \(\gamma \) . Hence, by unfolding the
procedure, the D-Log Padé \(\mathrm {Dlog}_N^M(z)\)  of \(f(z)\)  is
given by [1]}, [2]}
\(\mathrm {Dlog}_N^M(z) = f_{\mathrm {norm}}(0) \,\exp {\left\lbrace \int \mathrm {d}z^{\prime } \,\, \bar{P}_N^M(z^{\prime }) \right\rbrace } \; ,\) 
